
Here, \(w\)  is the equation of state parameter.
The solution can be obtained by means of Green's function.
We assume that the scalar-induced second-order gravitational waves are produced during the radiation-dominated era. The energy density parameter at the production time, \(\Omega _{\rm GW,f}(k) = \Omega _{\rm GW}(t_{\rm f},k)\) , is given by [1]}
\(&\Omega _{\rm GW,f}(k) =\frac{1}{12}\int _0^\infty dv\int _{|1-v|}^{1+v} du\left(\frac{4v^2 - (1+v^2-u^2)^2}{4uv}\right)^2\nonumber \\&\quad \times \mathcal {P}_\zeta (kv)\mathcal {P}_\zeta (ku)\left(\frac{3(u^2+v^2-3)}{4u^3v^3}\right)^2\nonumber \\&\quad \times \bigg [\left(-4uv + (u^2+v^2-3)\log \bigg \vert \frac{3-(u+v)^2}{3-(u-v)^2}\bigg \vert \right)^2\nonumber \\&\hspace{56.9055pt}+ \pi ^2(u^2+v^2-3)^2\theta (v+u-\sqrt{3})\bigg ]\,.\) 
